International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 19, Pages 1185-1192

On a thin set of integers involving the largest prime factor function

Jean-Marie De Koninck1 and Nicolas Doyon2

1Département de Mathématiques et de Statistique, Université Laval, Québec, Québec G1K 7P4, Canada
2Département de Mathématiques et de Statistique, Université de Montréal, Québec, Montréal H3C 3J7, Canada

Received 22 April 2002

Copyright © 2003 Jean-Marie De Koninck and Nicolas Doyon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For each integer n2, let P(n) denote its largest prime factor. Let S:={n2:n does not divide P(n)!} and S(x):=#{nx:nS}. Erdős (1991) conjectured that S is a set of zero density. This was proved by Kastanas (1994) who established that S(x)=O(x/logx). Recently, Akbik (1999) proved that S(x)=O(xexp{(1/4)logx}). In this paper, we show that S(x)=xexp{(2+o(1))×logxloglogx}. We also investigate small and large gaps among the elements of S and state some conjectures.